An algebraic expression is formed from letter symbols and numbers, combined with operation signs and brackets. Each part of an expression is called a term. In the expression 3n + 5 the terms are 3n and 5.

A formula is an equation linking sets of physical variables. For example, in the formula v = u + at, has 4 variables v, u, a and t related by the formula. If the values of three variables are known, the fourth value can be calculated.

An equation is a mathematical statement showing that two expressions have equal value. The expressions are linked with the symbol =. For example, in the equation 5x + 4 = 2x + 31, x is a particular unknown number for which the expressions on either side of the equation have equal value. Solving the equation simply means finding the value of x for which this occurs (in this case, x = 9).

In an identity, the expressions on each side of an equation always take the same value, whatever numbers are substituted for the letters; the expressions are said to be identically equal. The expressions are linked with the symbol ≡. For example, 4(a + 1) ≡ 4a + 4 is an identity, because the expressions 4(a + 1) and 4a + 4 always have the same value, whatever value a takes.

Additional User Example

Expression

5x +3

Identity

3x+9 = 3(x+3)

Formula

P=2l +2w

What this might look like in the classroom

Problem
In the diagram below, x, y and z are the lengths of the sides of the cuboid.
Use the diagram to find an (i) expression, (ii) formula, (iii) equation

Possible solutions (there are many)
2y + 2z is an expression with two terms that represents the perimeter of the shaded rectangle. This can be written as the formula P = 2y + 2z
where P is the perimeter.

What would be the formula for the perimeter of the other faces you can see?

A formula for the volume of this shape (V) would be V = xyz (where xyz means x times y times z).

What would the value of the formula be if x = 3cm, y = 4cm and z = 8cm? (V = 3 × 4 × 8 = 96cm^{3})

If we know that V = 60cm^{3} and that y = 3cm and z = 2cm then we know that 6x = 60. This is an equation, and we can work out that for this to be true there is only one possible value for x which is 10cm

Show that if the volume was 90cm^{3} and y and z were both 3cm, then x would again be 10cm

*****

Terms either side of an identity sign are always equal regardless of the value of the variables. A simple example of this would be that 4a + 2 = a + a + a + a + 2. We could take this further by saying that
4a + 2 = a + a + a + a + 2 = 4(a+1) − 2 = 2(2a − 1) + 4

Taking this mathematics further

Write down as many formulae as you can remember. Which formulae have 3 variables, 4 variables etc?

Find out what is meant by dependent and independent variables.

Making connections

Learners will begin to develop algebraic reasoning without even realising that that is what they are doing. The ability to pattern−spot and generalise in simple cases is well−established in the primary phase, and a clear understanding of the technical vocabulary helps ensure learners do not get confused as the process becomes more formal.

These ideas form a basis on which to build algebraic processes. The next stage would be to generate expressions, manipulate formulae and solve equations.

The equation in the example is a linear equation but the principle is the same for higher order equations e.g. those with terms in x^{2}, x^{3} etc. although there is then usually more than one value that works.

Learners sometimes find it difficult to understand the difference between an expression and an equation.